This is unfortunate for at least one author who I'll leave anonymous, because it provides a counterexample to a theorem in one of his books on one relator groups.
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JeremyKunApr 16 '11 at 3:31

Note that my answer only works when $G$ is in fact an HNN-extension of a free group, which isn't obviously the case for all groups with the kind of presentation you've given. (For instance, the relations could be $t^{-1}x_i t = x_1$ for all $i$, and so $t$ isn't giving you an isomorphism between subgroups.) Could that be the source of the problem?
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Richard KentApr 16 '11 at 4:19

No. It is definitely an HNN extension of a free group. I just posed the more general question because I didn't know of this theorem on HNN extensions, and thought to answer it from a different angle.
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JeremyKunApr 16 '11 at 16:26